This study is concerned with the natural action (as Möbius transformations) of some subgroups of PGL2(Z) on the elements of quadratic number field over the rational numbers. We start with two groups- the full modular group G = PSL2(Z) and another group of Möbius transformations M= 〈x,y: x^2= y^6=1〉. We consider different sets of numbers with fixed discriminants in the quadratic field and look at structure of the orbits orbits of the actions of G, M, G∩M and their subgroups on these sets. The results of earlier studies on the number of orbits and the properties of elements belonging to them are extended by similar results related to the new twist connected to the group M which has nontrivial intersection with G and opens a possibility to look at orbits which were not computed in earlier studies.